How to solve bessel's differntial equations

[manjunathan]

Bessel's equation, order 1/2 is given by:
(x^2)y'' + xy' + ((x^2) - 1/4)y = 0.
One solution is y=(sinx)x^(-1/2).

 

[HallsofIvy]

You are asking for another, independent, solution? The standard "reduction of order" method reduces this to a first order equation: Let \(\displaystyle y= x^{1/2}sin(x)u(x)\). Put that into the given equation and you get a differential equation with u'' and u' but no u. Let v= u' and you have a first order equation for v.